What rope would be most helpful/versatile in a survival situation?
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Most survival experts recommend parachute cord. From Wikipedia:
I'd argue that it's more than just a survival rope -- it's useful in many regular camp tasks like hanging a bear bag or putting up a clothes line.
Unfortunately, the term "parachute cord" has been slapped on cheaper knock-offs that aren't up to the military's original specifications, so it's not as strong. If you're going to purchase some, make sure you select a type that's strong enough for your intended uses.
As Hartley Brody suggests, I would carry a few metres of para-cord, as this is a very tough and versatile rope which can be used for a lot of things in survival situations.
However, if you are heading up onto a hill, I would also carry a 30m or 60m length of abseiling or climbing rope (and a belay device), just in case you ever need to make an emergency descent.
I agree that good quality para cord (look for 550 cord from a reputable supplier) is the best general purpose solution.
It is strong enough that you could just about abseil off it with nothing else than a couple of carabiners in an absolute life or death situation or use it as a safety line for crossing a river or lowering your pack over a steep incline or canyon wall etc.
It also has a multitude of other uses from bootlaces to cordage for shelter building and you can separate the inner threads to use as fishing line or sewing thread for repairs.
You can also twist or braid it together to make a stronger rope or straps for an improvised pack.
It is light enough that you can stuff a few hundred feet of it into a pocket and forget about it.
For survival use it is probably best to go for high visibility orange, yellow or red that way if you are waiting for rescue in dense terrain you can string it across trails or open spaces to guide rescuers to your location.
It should go without saying that if you are doing any sort of climbing, abseiling etc in anything other than a dire emergency you really do need to right tools and knowledge for the situation in hand.
It is also worth adding that a knowledge of the fundamental principals of cordage, ropes and knots is one of the most basis survival skills.
I may be biased as a mariner, but I am partial to three stranded twisted rope. It may not be as strong as braided-cored ropes per thickness, but I can do more with it.
With three stranded twisted rope, I can indefinately join pieces together permanently without any special tools. So I can make a continuous and very long rope if necessary that can still run through makeshift pulleys, sheaves and and blocks.
I can if necessary divide the rope up into its individual yarns if I need something smaller, and I can retwist them back together if necessary (admittedly a pain in the arse, but still doable).
I can put a permanent eye at the end of the rope without using a bowline or needing special tools to do so, for in those cases when a knot isn't right for the job.
If my rope starts to fray, I can cut it and rejoin it permanently without special tools.
Some specially knots, such as those to make animal traps from bent over trees take advantage of the knuckles caused by being twisted stands of rope, creating a knot that pulls free, but resists it until the knuckle gets pulled through.